This is the first part of a work whose goal is study associator equations in a way which is adapted to the framework of crystalline pro-unipotent fundamental groupoids. Our general goal is to reformulate (some natural consequences of) the associator equations as an explicit comparison between the respective modules of coefficients of an associator and its image under a certain automorphism, this comparison being compatible with their respective depth filtrations and defined over a ring of rational coefficients whose denominators have their $p$-adic norms bounded in a certain specific way. In this first paper, we achieve this goal for a certain part of the associator equations. We deduce from our result new proofs to known properties of "depth reduction" for multiple zeta values, i.e. the vanishing of certain depth-graded multiple zeta values ; this shows that associator equations can be adapted to the study of depth-graded multiple zeta values. One of these depth reductions has a specific application to $p$-adic multiple zeta values, which involves finite multiple zeta values, and the notion of adjoint multiple zeta values which we introduced in \cite{J2} and which we used for studying $p$-adic multiple zeta values. We also interpret others of our results and our approach in terms of the crystalline pro-unipotent fundamental groupoid of $\mathbb{P}^{1} - \{0,1,\infty\}$.